All of the 3rd grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$4.50$ each for students, and the group paid $$57.00$ in total. The next month, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$8.00$ each for students, and the group paid $$108.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4.5y = 57}$ ${15x+8y = 108}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-15x-9y = -114}$ ${15x+8y = 108}$ Add the top and bottom equations together. $ -y = -6 $ $ y = \dfrac{-6}{-1}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7.5x+4.5y = 57}$ to find $x$ ${7.5x + 4.5}{(6)}{= 57}$ $7.5x+27 = 57$ $7.5x = 30$ $x = \dfrac{30}{7.5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {15x+8y = 108}$ and get the same answer for $x$ ${15x + 8}{(6)}{= 108}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.